Epitrochoids

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Epitrochoids
Epitrochoid.PNG
Field: Geometry
Image Created By: Kevin Liu

Epitrochoids

An epitrochoid is a roulette made from a circle going around another circle. A roulette is a curve that is created by tracing a point attached to a rolling figure.


Basic Description

Epitroc4.gif

The basic concept of this is: a circle that can't move and a circle that can. The circle that can't move is fixed onto a surface (like paper). Within the circle that can move is a line that will create a line once this circle is moving around the fixed circle. Think of it as a rabbit that runs around its hole (that's hopefully on the ground) holding a marker while staying in the same position, but twirling.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Roulettes

Deriving the Parametric

1: First, it should be known that this is plotted in accordance with a po [...]

Deriving the Parametric

1: First, it should be known that this is plotted in accordance with a polar grid. So, let's just say that "m" will be the sum of the 2 radii from the 2 circles -> m=A+B

2: The equation for a circle with its center at the origin is always x^2+y^2=r^2, where r is the radius. Thus, for the equation of the larger circle that has the radius "A" and its center at the origin of a graph, it will be this. -> x^2+y^2=A^2

3: For the equation of a circle whose center is not on the origin, then the formula becomes (x-h)^2 + (y-k)^2 =r^2. Thus, the equation of the smaller circle (with radius "B"), its "X" value will have to be(x-m) in this case because to find the center of the smaller circle on the graph, you would subtract "m" (the sum of the 2 radii) from x. -> (x-m)^2 + y^2= B^2

4: Define point P on its distance away from the origin in coordinate form when t=0 -> P[0] = ( m-h , 0 )

5: Since the small circle (radius "B")revolves counter-clockwise around the the big circle -> P= m [cos(t), sin(t)] - h [cos( B), sin( B)]

sketch1.gif sketch22.gif

6: Angle B needs to be expressed in terms of "t", the angle of the big circle. Thus when the smaller circle rolls on the big circle, it produces the same arc length as the big circle.-> arc BC = arc RC

7: Now, because s = r(theta), "s" being the arc length, "r" being the radius and "theta" is the central angle in radians. -> A t = B t[1]

8: Solve the equation for t[1] -> t[1]= (At/B)

9:From the diagram, it's seen that -> B = t[1]+ t

10:Now substitute "B" into the equation from Step #8, to show Angle B in terms of "t"-> B = (At/B)+t

11:Since m=A+B, you simplify to get-> B = mt/B

Derivation from; http://msemac.redwoods.edu/~darnold/math50c/CalcProj/Sp99/Fran/epitrochoid.htm

Parametric Equations for an epitrochoid: epitrochoidparam.gif

In the explanation above, "A" corresponds to "a" in the equation and so on...

Special Epitrochoids

Limicon: When the value of "a" is equivalent to the value of "b" Epicyloid: When the value of "h" is equivalent to the value of "b"


Equations

(These equations were found at http://mathworld.wolfram.com/Epitrochoid.html)

This is the equation for the velocity of the rotating circle:

epitrochoidvel.gif


This is for the acceleration of the rotating circle:

epitrochoidaccl.gif


The Equation for the Arc Length:

epitrochoidAL.gif


This is the equation for the amount of curvature:

epitrochoidkap.gif


For all of these above equations: h is the distance from the point being traced to the center of the rotating/rolling circle.

The radius of the rolling circle is b.

The radius of the stationary circle, that the smaller circle is rolling around, is a.

Explanation on the Position of the Point

The amount of times the point being traced goes back to where it started on the revolving circle is based on the two different circumferences. If the circumference of the revolving circle is a, and the stationary circle is 2a, that means the point on the revolving circle will end up where it started twice (2a/a=2). In other words, it takes the rotating circle two full rotations in order to return to the place where it first started.

Different points traced.png

Where the point is does not affect how many times arcs will be traced, as long as it is attached to the revolving circle. As demonstrated by this image, there are 2 curves no matter where the point is. If the point is on the edge of the circle, the curves will touch the edge of the stationary circle (red). If the point is inside the rotating circle, the curves will not touch the edge of the stationary circle (blue). If the point is outside of the rotating circle, the curves will loop into the stationary circle (purple).

However, the positioning of the point being traced does affect how curved the line is when the point goes back to its starting point. right


Why It's Interesting

It creates interesting shapes and can be applied to uses of art and engineering. Two examples of usages of an epitrochoids include the Wankel Rotary Engine and the Spirograph.

Wankel Rotary

Wankel Cycle anim en.gif

The Wankel Rotary Engine was created by Felix Wankel in the 1920s that is used in cars and vehicles as engines. At the time, it was a new type of gasoline engine. The rotor itself is an equilateral triangle while the bore is made of an eptitrochoid curve. By following the path that the rotor tip creates, you can see the traces produced. For another explanation, you can go on: Involute

Url-2.jpeg

Spirograph

The Spirograph is a geometric tool used in manually drawing roulettes. It can produce epitrochoids. It can also produce hypocycloids, hypotrochoids, and epicycloid.

How the Main Image Relates

The concept of epitrochoids is explained.

Teaching Materials

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About the Creator of this Image

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Related Links

Additional Resources

http://mathworld.wolfram.com/Epitrochoid.html* http://anna466-epitrochoid.freeservers.com/history_n_uses.html






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